Adjoint of an operator

Poll without page-refresh

Article on other languages:

	del.icio.us
	Digg
	Furl
	Reddit
	Rojo
	Add to OnlyWire

In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex conjugate of a complex number.

The adjoint of an operator A is also sometimes called the Hermitian adjoint (after Charles Hermite) of A and is denoted by A^* or A^† (the latter especially when used in conjunction with the bra-ket notation).

1 Definition for bounded operators
2 Properties
3 Hermitian operators
4 Adjoints of unbounded operators
5 Other adjoints
6 See also

Definition for bounded operators

Suppose H is a Hilbert space, with inner product $\langle\cdot,\cdot\rangle$ . Consider a continuous linear operator A : H → H (this is the same as a bounded operator).

Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : H → H with the following property:

$\lang Ax , y \rang = \lang x , A^* y \rang \quad \mbox{for all } x,y\in H.$

This operator A* is the adjoint of A.

This can be seen as a generalization of the conjugate transpose or adjoint matrix of a square matrix which has a similar property involving the standard complex inner product.

Properties

Immediate properties:

A** = A
If A is invertible, so is A*. Then, (A*)⁻¹ = (A⁻¹)*
(A + B)* = A* + B*
(λA)* = λ* A*, where λ* denotes the complex conjugate of the complex number λ
(AB)* = B* A*

If we define the operator norm of A by

$\| A \| _{op} := \sup \{ \|Ax \| : \| x \| \le 1 \}$

then

$\| A^* \| _{op} = \| A \| _{op}$ .

Moreover,

$\| A^* A \| _{op} = \| A \| _{op}^2$

The set of bounded linear operators on a Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C* algebra.

The relationship between the image of $A$ and the kernel of its adjoint is given by:

$\ker A^* = \left( \operatorname{im}\ A \right)^\bot$

$\left( \ker A^* \right)^\bot = \overline{\operatorname{im}\ A}$

Proof of the first equation:

$\begin{align} A^* x = 0 &\iff \langle A^*x,y \rangle = 0 \quad \forall y \in H \\ &\iff \langle x,Ay \rangle = 0 \quad \forall y \in H \\ &\iff x\ \bot \ \operatorname{im}\ A \end{align}$

The second equation follows from the first by taking the orthogonal space on both sides. Note that in general, the image need not be closed, but the kernel of a continuous operator always is.

Hermitian operators

A bounded operator A : H → H is called Hermitian or self-adjoint if

A = A*

which is equivalent to

$\lang Ax , y \rang = \lang x , A y \rang \mbox{ for all } x,y\in H.$

In some sense, these operators play the role of the real numbers (being equal to their own "complex conjugate"). They serve as the model of real-valued observables in quantum mechanics. See the article on self-adjoint operators for a full treatment.

Adjoints of unbounded operators

Many operators of importance are not continuous and are only defined on a subspace of a Hilbert space. In this situation, one may still define an adjoint, as is explained in the article on self-adjoint operators.

Other adjoints

The equation

$\lang Ax , y \rang = \lang x , A^* y \rang$

is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name.